A bornology on a nonempty set X is a family of subsets of X that is closed under taking finite unions, that is hereditary, and that forms a cover of X. Bornologies have been widely applied in functional analysis and topology to form the general framework in the theory of locally convex spaces and to provide an axiomatic approach to boundedness in topology. Recently, there has been renewed interest in bornologies in topology, mainly stemming from hyperspace theory. In this talk, I shall present some recent results of mine and others on hyperspaces generated by various bornologies. I shall also discuss some open problems in this direction.